"Accepted" means the return was received in proper format by the IRS or state so they can start processing it.  It is the status of the efile, not the status of the refund.  I'll tell you how to check on your state refund below.

First, double-check with this tool to be sure your efiled state return was accepted:
https://shop.turbotax.intuit.com/efile/efile_status_lookup.jsp

Some states are quick with refunds, while some like WV or GA can take a long time, even several weeks.   There may be a notice or FAQ on your state website that tells about how long it takes for a typical refund in your state.

FAQ:  How do I track my state refund?