Hmm, "easy" is a relative term, and I tend to err on the side of too much detail, but bear with me here.
For Lydian scale, there is indeed a relatively simple way, since it uses the same diatonic pattern as the major and minor scales, just shifted. Consequently, the natural chords in the key exactly match the set for the related major and minor keys, just associated with different scale degrees. For instance, F Lydian is a relative mode of C major, and its scale includes the same set of seven pitches: (C D E) F G A B C D E F (...). Therefore, the I chord in Lydian mode matches the IV chord of major mode: they're both major triads, which extend into maj7 seventh chords. Here's how all seven Lydian degrees map out compared to the major scale, with the associated triad and 7th types for each degree:
Code:
Lydian Major Triad Seventh
I IV major maj7
II V major 7
III VI minor m7
IV VII mb5 ø = m7b5 (mb5 = the diminished triad)
V I major maj7
VI II minor m7
VII III minor m7
To figure out the key signature for a Lydian scale, either go down a fourth (five semitones) or up a fifth (seven semitones). The Lydian scale will share the same key signature as the major scale built on that note. For instance, a fifth below D is A, so D Lydian has the same key signature as A major: 3 sharps (F#, C#, G#). Consequently, you'd expect these diatonic chords in D Lydian:
I = D[maj7]
II = E[7]
III = F#m[7]
IV = G#mb5 or G#ø
V = A[maj7]
VI = Bm[7]
VII = Cm[7]
Of course, as we know from major and minor scales, other chord qualities may be substituted for the ones above; particularly, we might use V7 in place of Vmaj7 to increase the pull to the I chord, or for the IV degree we may either use a less dissonant chord type or substitute the more dissonant and ambiguous dim7 for the natural 7th m7b5 (the "half-diminished" chord, also notated "ø"). The degree matching the diminished triad is usually a bugaboo for the various diatonic modes. In Lydian mode, the IV chord, usually one of the central chords in a key, gets mapped to the diminished triad, so II is often substituted for IV instead. (In major mode, II is classed as a "subdominant" type of chord, along with the true subdominant, IV, so even in major mode you find these two chords substituted for each other in function.)
You'll find similar correspondences between the other "standard" diatonic modes and major mode:
Dorian: I corresponds to the relative major's II, etc. The relative major tonic is a whole step below, so G Dorian shares the same key signature and chord set with F major.
Phrygian: I corresponds to major III, etc.; the relative major tonic is a major third (four semitones) below.
Mixolydian: I corresponds to major V, etc.; the relative major tonic is a fourth (five semitones) above.
Aeolian (= natural minor): I corresponds to major VI: the relative major tonic is a minor third (three semitones) above.
Locrian (rarely used): I corresponds to major VII, etc.; the relative major tonic is a half-step above.
Let me also warn you that some pundits with a superficial understanding of tonality, modes and harmonic development will tell you "Lydian harmony is the same as major harmony, but you start the scale in the middle—it's all the same key." Don't buy it. This view disregards the many defining features of modes, such as tonic departure and return (both melodically and harmonically), frequency of chord usage, common progressions, common substitutions and alterations (as mentioned above) and the many other behavioral idiosyncrasies that have crept in over generations. So while identifying the natural, shared diatonic chord set is an excellent and necessary starting point, actual Lydian harmony goes much deeper and significantly departs from major harmony. The same applies to every other mode. The "all the same key" view may have held when modes first came into existence, but is nothing but a historical footnote in relation to modern practice, and has been for centuries.
Harmony for scales like Arabic, Hungarian and Indian ones (or the blues scales) gets complicated because they may contain scale steps larger than a whole step, and some third intervals that result when you skip over notes in the scale pattern have sizes larger or smaller than the minor and major thirds used in the "standard" diatonic pattern. When that happens, the derived triads can be either dissonant or modally ambiguous (having no minor or major third). In practice, the underlying harmonies can be regularized by substituting notes not in the scales, but as these substitutions are arbitrary, you end up with a richer assortment of harmonic alternatives—as indeed we see in the regular minor mode, where the treatment of the 6th and 7th scale degrees is rather fluid. Another complication is that some scales may include more than seven pitches (ignoring the octave)—how, then, do you build up the chords? Of course, one solution is to use an approach similar to "power chords": you don't always need a full triad or seventh. With altered scales like these, the focus tends to be on rich melodic and rhythmic improvisation, not on complex harmonic backing.
A similar thing happens in gapped scales (pentatonic and hexatonic, meaning they have five or six notes only): for harmonies, you often have to fill the gaps somehow, but exactly how you do it is arbitrary, and can change upon each instance.
So as you see, you've asked a very good question, but there is no simple answer except with relative modes of major and minor, like Lydian.
Interestingly, at the other extreme we have the diminished and whole-tone scales. With the two diminished scale patterns, which alternate whole and half steps (for instance, C Db Eb E F# G A Bb C and C D Eb F Gb Ab A B C), every third interval is a minor third, so the derived triads on every degree are diminished triads, while all the sevenths are diminished sevenths. Similarly, all the thirds in the whole tone scale (like C D E F# G# A# C) are major thirds, so all the triads are augmented triads and all the seventh intervals would be unisons with the octave, devolving the seventh chords into augmented triads as well. Sticking to these natural chords would be boring indeed—though when interjected into songs in other scales, diminished or whole-tone scale passages can be quite effective.
