If you buy a bond and hold it until maturity, then you really wouldn't feel any interest-rate risk.
The risk only becomes real if you hold a bond which you intend to sell before it matures.
This is why the yield curve slopes upward, generally: to encourage you to buy bonds which have a longer maturity, even if you plan to get your money out earlier. In this sense it is a type of risk premium.
A bit more on that.
Suppose you want to invest your savings for a couple of years, and see that you could buy government bonds with a maturity of 2 years which have an implied yield of 2%. This is appealing, since you will definitely (barring a government default) have your savings back intact at the end of year 2.
Now your broker tells you that the 10-year government bonds are yielding 3%, "so why don't you buy the 10-year bond, get the extra 1% interest for a couple of years, then sell the bonds back into the market?" he suggests.
Now you need to have the 'interest-rate risk here!' alarm bells ringing.
Let's walk through the calculations. If you invest $100 in the 2-year bond, you'll get a couple of coupons and your $100 back at the end of 2 years: say $104 in total cash.
Alternatively, if you invest your $100 in the 10-year bond, then at the end of year 2 you will have received 2 coupons worth $6 in total, but you won't get your initial $100 back because the bond has not matured yet. Instead, you are the owner of an 8-year government bond with a coupon of 3%.
Cutting to the chase, if interest rates move dramatically higher over the next 2 years, then (when you get there) that 8-year government bond will be worth less than $100; so although you had higher coupons from the 10-year bond, you probably won't be able to sell the 8-year bond back to the market at a price near $100. All things considered (coupons and capital) your cash position will be worse than $104 - you should have not been distracted by the higher coupon of the 10-year bond.
Interest-rate risk only matters if you do not hold to maturity.
If you don't plan to hold a bond to maturity, then you need to be comfortable that the extra yield you get is enough that you feel it comfortably compensates you for the risk that interest rates move upward and damage the price at which you would sell your bond when you want to take your money out of the investment.
(I covered this in my blog, btw: www.fermatslastspreadsheet.com)