How can I cast an NSMutableArray to a Swift array of a specific type?

You can make this work with a double downcast, first to NSArray, then to [String]:

var strings = myObjcObject.getStrings() as NSArray as [String]

Tested in a Playground with:

import Foundation

var objCMutableArray = NSMutableArray(array: ["a", "b", "c"])
var swiftArray = objCMutableArray as NSArray as [String]

Update:
In later versions of Swift (at least 1.2), the compiler will complain about as [String]. Instead you should use an if let with a conditional downcast as?:

import Foundation

var objCMutableArray = NSMutableArray(array: ["a", "b", "c"])
if let swiftArray = objCMutableArray as NSArray as? [String] {
    // Use swiftArray here
}

If you are absolutely sure that your NSMutableArray can be cast to [String], then you can use as! instead (but you probably shouldn't use this in most cases):

import Foundation

var objCMutableArray = NSMutableArray(array: ["a", "b", "c"])
var swiftArray = objCMutableArray as NSArray as! [String]