Privacy Fence Rails alternating idea

I am building a 6' privacy fence and wanted some feedback. I know everyone butts their 2x4 joints in the middle of the post. I was thinking of installing the rails in an alternating pattern so that each 2x4 rail covers the entire post instead of just half of the post width; rails over/under each other all the way down the fence line. I am just not sure if it will look strange, normally when you look down a fence line, the rails are all straight the whole length. Seems easier to build (not having to measure and cut between each post, can cut ends off after with a sawsall) and stronger and more of the 2x4 is in contact with each post. Thoughts, Ideas? Thanks.