A new cultivar of Ulmus tree named ‘;DavesStraightUp’; that is characterized by its upright and very columnar plant habit, its small leaves, its branch tips that are upright and do not splay with age, its strong crotch angles that are tight and average between 20° and 25° to vertical, its exfoliating bark that tends to curl in strips and reveals colorful patches of orange-tan, grey and white, its fall foliage that turns to yellow and yellow-orange in color and lasts late into the season, and its resistance to Dutch elm disease and phloem neurosis.