Azure files have Shared Access Signatures (SAS). This is a key that you compute with the storage account key, that gives access to a particular URL. Here is an example (storage account name is obfuscated here):
https://mystorageaccount.file.core.windows.net/sampleshare/2.png?sv=2015-04-05&sr=f&si=sampleread&sig=Zq%2BfflhhbAU4CkCuz9q%2BnUFEM%2Fsg2PbXe3L4MeCC9Bo%3D&sip=0.0.0.0-255.255.255.255
You have sample code on how to create a SAS with Azure files at https://azure.microsoft.com/en-us/documentation/articles/storage-dotnet-how-to-use-files/, §"Generate a shared access signature for a file or file share".
You can also do it interactively with a number of tools. For instance, CloudXPlorer has this feature.