You could use CallContext.LogicalSetData and CallContext.LogicalGetData, but I recommend you don’t because they don’t support any kind of “cloning” when you use simple parallelism (Task.WhenAny / Task.WhenAll).
I opened a UserVoice request for a more complete async-compatible “context”, explained in more detail in an MSDN forum post. It does not seem possible to build one ourselves. Jon Skeet has a good blog entry on the subject.
So, I recommend you use argument, lambda closures, or the members of the local instance (this), as Marc described.
And yes, OperationContext.Current is not preserved across awaits.
Update: .NET 4.5 does support Logical[Get|Set]Data in async code. Details on my blog.