I am looking for an FDIC (or other reliable body) insured account that pays more in interest than CDs. What is the next step up, interest wise, from CDs, even if it means tieing up the money longer and / or other restrictions?
You don't mention how much money you are talking about but one option is to use reward checking accounts that are FDIC/NCUA insured. They pay 3-4% interest but generally have a few requirements such as 10-12 debit card transactions and sometimes require direct deposit as well as a limit of 10-50k deposits earning the top rate.
Realistically, it is CDs with longer terms or are callable.
You pretty much have to accept more risk if you want higher returns. If you are willing to accept that risk by losing the FDIC protections the next level up is probably high rated Government bonds.
FDIC insurance only protects certain funds, and is not meant to protect against losses in the market, or to guarantee every investment you might make.
Generally, the only accounts that will be FDIC insured are savings, checking, money market and CDs.
There is no such animal.
If you are looking to give up FDIC protection, investing in a short-term, high quality bond fund or a tax-free bond fund with short durations is a good way to balance safety vs. return.
Make sure you buy funds -- buying individual bonds isn't appropriate for folks without a high net worth.
Another option is savings bonds, but the yields on these is awful today.