The get/put/post calls have been streamlined to require fewer parameters in the basic case - i.e. you can just call http.get("www.mysite.com", "/somepath") to make a simple request.
The accept header has been removed from the list of possible parameters to get/put/post - if you need to use it then send it manually with sendHeader(...) instead.
You don't need to call finishRequest() after the initial call to get/put/post if you aren't going to send any headers. However, if you /do/ want to send extra headers then you need to call beginRequest() before the get/put/post and endRequest() at the end of all the sent data (so after the data as well as the headers). E.g.
http.beginRequest();
http.post("www.somesite.com", "/somepath");
http.sendHeader("Content-Length", strlen(postdata));
http.print(postdata);
http.endRequest();