For portfolios comprised of instruments in the U.S., Britain or other countries with fairly low credit risk to the government, this is traditionally done by trading various maturities of treasury bonds.
A simple technique is to divide your portfolio instruments into "buckets" of duration, say 0-2, 2-5, 5-10, and 10+ years. Then, you sum up the exposure in each bucket, and hedge with opposite exposure to treasury bonds of approximately the same duration.
For a more sophisticated approach, you can create a factor model, usually with $N=3$ factors, and compute exposures and hedges using that. But doing so is somewhat more complicated than the bucket approach without necessarily performing better.
If treasuries are impractical for some reason, high-grade corporate debt can serve as an acceptable proxy. Some organizations are starting to use ETFs as well, but of course they are composite instruments which makes them tricky to handle on traditional curve models.
In derivatives land, you might hedge with interest rate fixed-for-float swaps. These are standardized and nearly as liquid as treasuries, while also being nearly as easy to work out exposures with. The main trouble is that using them involves setting up more sophisticated trading relationships with your counterparties, involving agreeing on (and signing) an ISDA, etc.