I'll advise you on the basis of our
experience in our greenhouse, rather than a broad overview of many different greenhouse situations. Our G.H. is an organic & fairly deep-soil situation, with a soaker hose
irrigation system that we rely on for the most part.
We've used clipped grass mostly - green grass, clipped (e.g., by scythe, weedwhacker,
lawn mower) while not too long
and before viable seeds have matured in it. We've put on a layer of maybe an inch thick. You're right to be concerned about mold, I think - but in our experience keeping the mulch about and inch or a bit more away from the shoots or stems emerging up from the soil is
enough. Molds seem to take hold on main stems just above soil level. So it's been fine to have the mulch above the
roots. When the grass has gone yellowish or brownish and begun to decay into the topmost soil layer, we add some more grass to keep the layer about an inch thick.
This retains moisture partly by absorbing water, partly by shading the soil from direct sunlight, and partly by protecting the soil from warm air currents in the greenhouse. It nourishes the soil with natural bacteria.
Works for us in our specific locale and greenhouse.
