You could actually have MORE then 3 20 amp circuits. You just cant exceed the total of the sub panel feed at any one time, which would happen to be 60 amps. That probably doesn't make much sense... But if you look at a typical main panel on a house it maybe a 150, or a 200 amp panel. But if you add up all the ampacity of the individual branch circuits it will likely total MORE then 150 or 200 amps.
I have a 200 amp main and coming off of that panel I have a 100 amp sub panel. I also have over 10- 20 amp branch circuits for lighting and convenience outlets. Plus I have a 30 amp hot water heater, A/C circuit, ect. So WAY over 200 amps in theory, but in practice I won't be maxing out all those circuits at the same time. So I don't trip my main.
So you could have for instance 6- 20 amp circuits running to your room for all kinds of things like lights, fans, A/C, ect. You just can't max out any individual 20 amp circuit or in total the 60 amp feeder. There are many calculations to consider. For instance, you don't really want to continuously use more then 80% of a branch circuits capacity. So on a 20 amp circuit, you don't want to be using more then 16 amps continuously. So if you actually need a full 20 amp continuously, for different appliances you would run 2- 20 amp circuits to cover your needs. So now you have 40 amps worth of branch circuits even though you are only using 20 amps.
Just one problem... I was expecting you to have 2 hots, 1 neutral, and a ground. Sounds like you have 2 hots and the electrical box that the dry plug was in could be grounded as if the wires were run in conduit or something? Or is there a smaller bare wire screwed to the back of the outlet box? But that would still leave you without a neutral. At the dryer outlet, where did you connect the white wires from your smaller yellow romex cables you ran to the room?
Without seeing what you have, it sounds like you can't really have a 120/240 volt sub panel as I was thinking. There is a way to have just a 120 volt sub panel with the wires you have IF you truly have a GOOD ground at the dryer outlet. You could still have several 120 volt circuits to run the room, just no 240 volt appliances like some ballasts and big A/C's.
Best option, is run a whole new cable from the main panel to the grow room. Is that feasible? The problem is, now you are working in the main panel to get the new cable in. It was a much easier project when you didn't have to do any work in the main panel...
If running a new cable is not feasible, you can still explore using a 120 volt sub panel with the wire you have. But again, I have no idea what kind, or how good your ground is to the dryer outlet. The ground is a pretty important piece of the puzzle.