If I split a 240v 30amp (has neutral) circuit into two 20amp 120v circuits, would that count as two 20-amp branch circuits to meet code?
Yes, if you do it right. This is an MWBC, Multi Wire Branch Circuit.
The keys are:

240V double-breaker (or technically handle-tie is OK, but double-breaker is easier and you better already have that for a dryer anyway).
Feed must include hot/hot/neutral/ground. This might get tricky. A proper modern 30A dryer connection will have all 4 wires (or 3 + metal conduit for ground). However, older dryer connections only had 3 wires. You need to check this first. Upload a picture of the dryer receptacle and we can figure it out.
Replace the 30A breaker with a 20A breaker.
You need GFCI. That can either be done with a 20A double-breaker with GFCI built in (may or may not be available for your panel) or by using a GFCI receptacle as the first receptacle in each chain of receptacles in the kitchen.
The 30A receptacle goes away. Inside the box, run connect two 12/2 (12 AWG hot/neutral/ground) cables. (This could be done with a single 12/3, but that gets confusing and forces you to use breaker GFCI. Easier to avoid it.) The hot (black) wire from one cable goes to one hot wire from the panel. The hot (black) wire from the other cable goes to the other hot wire from the panel. The neutrals from both cables go to the neutral from the panel. All grounds (as always) together.
Each cable can now feed to as many receptacles (15A and/or 20A) as you want. All are ordinary receptacles except that if you did not install a GFCI breaker then the cable goes to the line side of a GFCI receptacle and all following receptacles go to the load side.