There are numerous papers on this topic, but put simply they need:

Light source (Plant growth lights are ideal)

Agitation to stay suspended (stir plate works)

Media (BG-11 is a common one for cyanos, though optimized media exist for each genus)

CO2 (Can be air or a CO2/air mix that is sparged in or a carbonate source like sodium bicarbonate)

Temperature Control (Room temp should be okay, but they will grow faster at slightly elevated temperatures like 30C)

Clear growth vessel (Like a jar works so that light can penetrate in)

In addition to the factors that Mr. Matthew has already reported, I suggest some punctual and more accurate information:

- ideal pH for most cyanobacteria is slightly alkaline (between 8.5-9.0)

- nitrogen level offered should prevent the lush phases

- if you have CO2 available, use a 0.3-0.5% enrichment

- Homogenization should be moderate for filamentous species.

I hope I was helpful.