I wrote C++ for 10 years. I encountered memory problems, but they could be fixed with a reasonable amount of effort.
For the last couple of years I've been writing C#. I find I still get lots of memory problems. They're difficult to diagnose and fix due to the non-determinancy, and because the C# philosophy is that you shouldn't have to worry about such things when you very definitely do.
One particular problem I find is that I have to explicitly dispose and cleanup everything in code. If I don't, then the memory profilers don't really help because there is so much chaff floating about you can't find a leak within all the data they're trying to show you. I wonder if I've got the wrong idea, or if the tool I've got isn't the best.
What kind of strategies and tools are useful for tackling memory leaks in .NET?
You still need to worry about memory when you are writing managed code unless your application is trivial. I will suggest two things: first, read CLR via C# because it will help you understand memory management in .NET. Second, learn to use a tool like CLRProfiler (Microsoft). This can give you an idea of what is causing your memory leak (e.g. you can take a look at your large object heap fragmentation)
I prefer dotmemory from Jetbrains
The best thing to keep in mind is to keep track of the references to your objects. It is very easy to end up with hanging references to objects that you don't care about anymore. If you are not going to use something anymore, get rid of it.
Get used to using a cache provider with sliding expirations, so that if something isn't referenced for a desired time window it is dereferenced and cleaned up. But if it is being accessed a lot it will say in memory.
From Visual Studio 2015 consider to use out of the box Memory Usage diagnostic tool to collect and analyze memory usage data.
The Memory Usage tool lets you take one or more snapshots of the managed and native memory heap to help understand the memory usage impact of object types.
Big guns - Debugging Tools for Windows
This is an amazing collection of tools. You can analyze both managed and unmanaged heaps with it and you can do it offline. This was very handy for debugging one of our ASP.NET applications that kept recycling due to memory overuse. I only had to create a full memory dump of living process running on production server, all analysis was done offline in WinDbg. (It turned out some developer was overusing in-memory Session storage.)
"If broken it is..." blog has very useful articles on the subject.
One of the best tools is using the Debugging Tools for Windows, and taking a memory dump of the process using adplus, then use windbg and the sos plugin to analyze the process memory, threads, and call stacks.
You can use this method for identifying problems on servers too, after installing the tools, share the directory, then connect to the share from the server using (net use) and either take a crash or hang dump of the process.
Then analyze offline.