I just went through this where the calcium buildup inside copper pipe was breaking loose.
The best solution I found was to:
1) Shutoff the water.
2) Remove aerators from all faucets.
3) Open the bathtub faucets.
4) Drain water from the bottom of water heater till it was clear.
5) Turn the water back on and flush the pipes.
6) I then disconnect the Washing Machine hoses at the machine, place them in the tub and turn on the water.
I recommend this as a separate operation unless you have a helper who can hold the hoses in the wash tub as the force of the water may cause the supply hoses to be pushed from the tub.
After doing this it just left stop (angle) valves at the toilets. In my case the calcium pieces were quite large and could not be flushed by just removing the toilet supply line at the tank and placing the hose end in a bucket and flushing. Since my angle values were compression fittings, I made up a jig using a combo female compression to female IPT and screwing a 3/8 inch barb fitting into the IPT side. I then removed the angle value, connected the compression end to the existing nut, connected a hose to the barb, placed the hose in the toilet and turned on the water. This did a great job of flushing the pipes at their endpoints. A word of caution regarding using the existing compression feral and nut. You need to compare the depth of the compression side of the combo against the depth of the compression side of existing angle value. If the combo has a deeper depth your good to go, otherwise you will need to redo the feral to keep from leaking at the compression fitting when the jig is installed.
FWIW I also made up a jig to flush the bathroom tub and shower faucets. Fortunately they were the original Price Pfister two handled so worked well. How one would make a jig for the cartridge type shower bodies to flush them I haven't had to deal with yet.
This tool works if you have the incoming pipe open and can use a garden hose to back flush the lines.