The original question deals with Windows 7, but it's happening now for a new reason in Windows 10. Here's a general answer that deals with both situations.
The general problem
There's an excellent explanation of the problem and its cause at the 2012 MSDN post Why is there sometimes a long delay between pressing a hotkey for a shortcut and opening the shortcut?
Executive summary: before launching a program via its shortcut key, Windows first polls all currently running programs and asks "Is this your shortcut key?" If so it switches focus to that window rather than spinning up a new copy of the program.
The problem happens when a window somewhere refuses to answer the question. This can be a program that's generally non-responsive for some reason, though it can also happen for other reasons (see below). Windows wait 3 second before muttering "jerk" under its breath and moving on to the next window to ask it the same shortcut question.
The general fix
If this is causing your delay, you can pretty easily find the cause.
- Start Windows, then before opening any other programs try a shortcut key. Do you experience a delay? If so, go to step 3.
- Launch any other programs you typically run. You should now experience the shortcut key delay. (If not, wait until you are experiencing it later before proceeding.)
- One by one, stop each of the programs. After stopping each one, see whether the shortcut delay has gone away. You need to stop not only programs running with actual windows, but also programs that appear only as icons in the taskbar's notification area (aka the "system tray"). You might need to expand the notification area to see all of those icons.
Now that you know your culprit, you get to decide: is that program worth the delay it causes? For example, Adobe Creative Cloud has sometimes caused problems for me, so I just tell it not to run when Windows starts. I can still use Illustrator etc. without it running.
If you can't live without the program, either live with the shortcut key delay or, as "user 99572 is fine" suggests, try a macro program like AutoHotKey.
Windows 10 Specific: Action Center
It seems that Windows 10 has introduced a twist, presumably causing this to happen for more people. (This might happen in Windows 8 too; I skipped from 7 to 10).
In the notification area you may see an icon like this:
This is the Windows Action Center telling you that it has system notifications for you to review. When this icon is in this state, it causes the 3-second shortcut delay (at least for me). The Action Center isn't non-responsive per se—it responds instantly when you click it. But apparently for some reason it doesn't respond to Windows shortcut key queries, causing the 3-second delay.
Click the icon and clear the notifications, then the icon will look like this:
This prevents it from causing the delay.
If a notification comes up repeatedly and you find it unhelpful, right click it and tell Windows to disable notices for that application.
Windows 10 Specific: Store and Calculator
Sometimes the delay happens even after doing all of the above. I've narrowed down the problem to two background processes:
Note that they're background processes: they don't appear as interactive windows, and in fact I haven't run them during the current session. For some reason one or the other just appears as a background process—seemingly at random. It's not clear yet why.
When I use task manager to kill them the shortcut delay goes away.
This 3-second timeout probably doesn't cause every keyboard shortcut delay.
One one of my laptops, whenever Adobe Creative Commons is running in background I sometimes experience a delay lasting as long as 30 seconds before launching an app. That's 10 times the timeout Window uses before giving up on a non-responsive process, so something else must be going on in this case.