We are going to Russia for 10 days. My husband will need to make many business calls to the US from there. What is the cheapest way to make international cell phone calls from Russia?

For Russia, this is it:
http://www.airbalticcard.com/

Difficult to reach (Estonian number), but free incoming , 0.41€ outgoing to US. Good quality outgoing calls in Russia to anywhere. Shipped to your door directly from Estonia.
Card is 20€ plus 9€ p&p with 10€ in calls.

In normal circumstances I recommend that clients get a local cell phone (MTS or Beeline, for instance) and use a calling card (Zebra). This is what my family does as well (we live in Moscow and the US).

This recommendation is for travelers to Moscow and St Petersburg; if you are going to another region I may have a different recommendation.

We have friends in Moscow who will lend us Russian cell phones. We are spending two days in Suzdul - do you think the 'Zebra' card will work there? and where can I buy one? Thanks for your help

You can buy a sim-card for international phone calls and put it into your mobile. I use one and it costs from 10 to 30 cents per minute to call outside Russia. The sim-card is 150RUB local price.

Thank you for all your help.

Zebra will work in most of Russia (they have local access numbers in most cities and regions).

Although I live in both the US and Russia, I am not aware of a SIM card only for international calls as described above.

I prefer Skype myself since it is so cheap, $0.021 per minute and because the voice channel quality is MUCH better than the dial around cards or even local phone land line or cell phone calls.

SIM card here are about $10-12 for a local phone number but be aware the phone needs to be GSM modulation and the Russian/European bands. If his current phone is not GSM and unlocked skip all the hassle of trying to get it unlocked and buy a used phone for $25-100 depending on model at a local phone store here because it will be unlocked so it can accept any GSM compatible SIM card. Any phone that works in Europe also will work here.

Roaming charges will be more than your hotel room so, a friend who had two SIM cards and mistakenly called me recently from just down the street ended up with a $132 charge for a 2 minute call. A long call would be horrendously expensive.
Bring his laptop and call from the many free Wi-Fi access points in cafes or hotels, and get a Skype Out and a Skype out phone number so people can call a local number there and have it ring where ever he is.

The second best way would be for outgoing only, using a local phone international phone card available at cell phone shops and get a local phone number SIM card. I use Megafon because it covers the country best with more cells.