I totally disagree with everyone who is shouting out "just use diminished 5ths", because a diminished fifth can sound smooth as silk in the right context with the tonality, while major 3rd or a pure fifth can sound like poison covered thorn in the the right context with the tonality.
The best way to sound weird or dissonant is to eliminate all feeling for tonality and the easiest way to that is to use:
Multitonality: Write for 4 voices, the first in C melodic minor, the second in E Octatonic (E-F-G -G#-A#-H-C#-D), 3rd in F# major and the 4th in 3rd Messiaen mode from Ab (Ab-B-H-C-D-D#-E-F#-G).
Or: Contrapuntally, use parallel 2nds, augm.4ths and 7ths and on odd numbered beats (heavy) always have one of those intervals between the bass and most audible voice. Harmonically, avoid the minor and major chords, you can use them, but only with minor second in the bass. Then use cords based on 4ths (with preferably one augmented) and 5ths (with preferably one diminished), and just plain odd chords, like the french one (major 3rd, major 2nd, major 3rd) and fully-diminished/augmented cords with a random added/shifted note, just be careful not to get any major/minor/dominant cords.