Hmmm Why did you get a google phone, You might have been better off with Casio G'one or a samsung if you wantted BING or something...But yes, Amazon AppStore and they give you a free app everyday..
there are apps like appbrain. you eventually are directed to the "market" for the actual download of the app....but the way it is set up (searching, filtering, etc) is different than the market. there's also a few developers where you can get the apps directly from their site, but that is WAY more work to deal with than just going to the market. i like appbrain. it's a handy app and has a website too.....but again, once you find the app you want through appbrain, it directs you to the market link to actually download it.
Does anyone out there have a reliable way to download apps besides through the "Market". The whole Google thing is getting a bit tiresome.
As others have mentioned, there is now available the Amazon apps store and also Appbrain for downloading apps.
A word of caution when using other methods of downloading apps other than from the Market...be aware that not all apps download are from trusted sources and can potentially cause problems on your phone.
In order to download from other than the market, you have to enable the "allow from unknown source" option in applications, which opens up your phone to the possibility of some type of attack.
Also as another poster mentioned, you say, "The whole Google thing is getting a bit tiresome."...Well I hate to tell you this but your will be forever tied to Google if you are using an Android device, since, well Android is Google and Google is Android.
Regardless of how or where you get your apps from, the process is still the same, so I'm not sure what is so "tiresome" about the way Google does it as opposed to anyone else?
Just my two cents.