Diagonal is not good - what about trying out cork? Cork floors come in tones that look great next to wood floors and they are terrific these days.

I thought of cork, but was discouraged because I have 2 80-pound golden retrievers. Not sure of the durability with these 2 guys running around all the time.

Assuming you have the traditional oak floor that is laid and finished in place, yes you can very successfully install additional oak floor next to it without matching issues. The key to this is that the whole floor ( old and new) has to be re-sanded and then re-finished. You may see just a slight variation in the new to old wood, but over time that fades as the new is exposed to UV. We do this all the time in our remodels with great success. Having all the same floor does enhance continuity in the house as opposed to going with another surface. With large dogs, a porcelain tile would hold up very well especially installed with an epoxy based grout. There are some beautiful tiles out there these days.

Unless you want to use tile or stone, I agree that matching is the best option, and Riddle's advice to have the same wood installed and then refinish both floors will pretty much guarantee the result. We even managed to match 90 year old white oak floors to new ones in an addition, and we didn't resand the old wood because it has gotten thin from a number of refinishings over the years. But a talented wood floor specialist managed to get a near-perfect match anyway. Old wood is on the top half of the picture; new wood is on the bottom half.