"I don't need to download new wallet software in the next 36 or so hours in order to take part, do I? "
No, if someone says you need to download some wallet to claim the BCC then it's a phishing scam.
If you hold your bitcoins in an address that you know the private keys then you will have controll over your BCC.

For now the wallets that will show you your BCC if you import the private keys are the wallets listed on the official bitcoin cash site.

The integration of BCC into more well known wallets will take some time, for example jaxx:
http://decentral.ca/jaxx-statement-bitcoin-cash-bch/

If you don't trust those wallets and don't want to wait for more well known wallets to implement BCC you could send your BTC to a trusted exchange that will credit you with the BCC, like bittrex.