The interest rate risk on a house isn't just the mortgage payment. It's also the value of the house itself.
House prices tend to move inversely to interest rates. When interest rates are low, then new home buyers can afford a bigger principal values on a loan of a given monthly payment. The market adjusts to accomodate this, and house prices rise to take up that principal value. When interest rates fall, new home buyers can't afford to take out as big a mortgage, so the supply of cash available in the market declines, and prices fall accordingly.
In 2006, when interest rates were at historic lows and home prices were at historic highs, I worked out who was making the profit from all of this. Obviously, existing homeowners who sold into that market were making a killing; where was the money coming from? It wasn't coming from the people who bought the house: they were paying the same monthly payment, for the same loan terms, as people who bought in the not-so-bubble years before. It might've been coming from the banks, but ultimately their profits were indifferent too, because they were just passing along their low borrowing costs. Trace the money all the way back, and it was being injected directly into the economy by the Fed, through low discount rates. That reduced the banks' borrowing costs, which reduced their mortgage rates, which increased the size of the loan that could be written for a given monthly payment, which increased the amount of cash in the housing market, which made housing prices go up.
And then I wondered what would happen when this system went into reverse, interest rates started to rise, and cash came out of the system. This started to happen in 2007, but then the system froze up, the Fed panicked, and the floodgates opened again. It will probably happen again at some point in the future, unless we get full-on hyperinflation.
Anyway, the obvious losers are people who bought houses with inflated mortgages for inflated prices. With high interest rates, new home buyers can't afford as big a mortgage for the same monthly payment, so house prices must come down for anyone to be able to buy. But who are the winners? Not (really) banks, who're just passing along the borrowing costs from the deposits they have. It's actually people who are holding cash right now and looking to buy in the near future. They're acting as a mini-bank in their own right: if you can buy a house with cash (or put down an absurd down payment like 50+%), then you're effectively acting as your own bank, but without any borrowing costs. Any rise in interest rates goes straight to your pocket.
tl;dr: Falling interest rates are good for homeowners, ambivalent for mortgaged buyers, and bad for people with cash savings. Rising interest rates a bad for homeowners, ambivalent for mortgaged buyers, and good for people with cash savings.