If MoneyGram money transfer, for some reason has not been received by the addressee, the sender will receive a refund. If you need to cancel or change a transaction, then you should contact by telephone with the MoneyGram branch agent, made the transaction for you. What are the main benefits of money transfers by MoneyGram? Cost. Today, it is most advantageous to the customer service of all the similar services. The benefits of money transfers by MoneyGram: the opportunity to send a telegram with money transfer (up to 10 words) for free. MoneyGram completed the rollout of simplified pricing for money transfer services in the U.S., an initiative that began last year.