There are several solutions depending on what you want to use instead of myetherwallet. A standard one would be to use geth and mist.
Once you will have geth installed, there should be an ethereum folder somewhere on your computer. On linux, it is probably
~/.ethereum. For geth to find your keys, you must copy them in the
Then geth and mist (using geth) should be able to use your keys.
It is a good practice for you to copy your myetherwallet
keystore/ folder on at least one device you own (even without installing geth): then if anything happens to the website (like a DOS attack) you still can spend them by installing geth or any other ethereum client.