You could achieve this by getting the transactions an address has been involved in and filtering on the outputs with that address hash.

We've just released our own free API called Blocktrail and provide SDKs for PHP, Python and NodeJS which make getting blockchain data trivial.

Assuming PHP you could do the following:

Initialise the Blocktrail SDK

use \Blocktrail\SDK\BlocktrailSDK;
$client = new BlocktrailSDK("MY_APIKEY", "MY_APISECRET", "BTC", false);

Get the address' transactions

$address = "1A4NfSJDzxzKgRrgfjxSAQ8hCPJJWKcmzX";
$page = 1;
$limit = 500;
$sortDir = "desc";
$transactions = $client->addressTransactions($address, $page, $limit, $sortDir);

For each transaction get the amount sent to this address from the outputs

$deposits = array();
foreach($transactions as $tx) {
    foreach($tx['outputs'] as $txout) {
      if($txout['address'] == $address) {
        $deposits[] = $txout['value'];
      }
    }
}

Tutorials and Examples

We'll be creating in-depth tutorials really soon, but for now you can check the API documentation for code examples and descriptions of the data returned.

There is also an example project on github: A Simple Block Explorer