tl;dr You're fine, you can fully utilize your processor and you will not have any troubles with performance. A new motherboard/ram is not required.
CPUs have a cache on them, this is where all data access takes place. If there is data that is in memory but not in the cache, it has to be loaded into the cache first.
The speed of the ram has less to do with how long it takes to access it than the memory timings do. These timing specify exactly how many cycles it takes to access the RAM. You can see this Wikipedia Article for details about memory timings.
As Far as your CPU is concerned, it actually has a internal clock speed much less than 2 GHz, what gives you the 2 GHz effective clock speed is the CPU Multiplier. As long as your base clock speed is less than the speed for your ram, you're fine. For example, my i5 2500k runs at 3.6GHz, its base clock speed is 100hz, and its multiplier is 36.
Another thing to be aware of, is that your ram isn't actually running at 1600hz, is running at 200hz. You can get a table of that info here. But as I said above, that 200hz is higher than the base clock speed of 100hz, so even on a good processor like my i5, 200hz is more than enough speed.