The Civil War Monument in Atkinson, New Hampshire is located at the town common in the center of town, on Rt 121 (Main Street). William Cleaves Todd, a native townsman and former teacher at the Atkinson Academy, gifted the granite memorial to the town in 1890. There used to be small granite pillars with iron railings surrounding it, but those have since been removed, probably when the common was redone (the pillars now mark the outside corners of the property).

On the four sides are inscribed the names of 40 Civil War soldiers who served from this town. With the town record books not having kept an accurate list of those who served, a list of names in the appendix of an 1875 historical speech about the town and Congregational Church (see link below) was used for reference.

Below are the names of the inscribed [additions/corrections in brackets]:

More About (with links):
Morse, Charles Fitch. Historical Discourse: Delivered at Atkinson, N.H., on the Centennial Anniversary of the Congregational Church. Lawrence, MA: Geo. S. Merrill & Crocker, c1875 [Appendix D, pg 36 - google books]
The Derry News, 31 Jan 31 1890 (monument gift, pg 5)
The Derry News, 31 May 1901 (article about monument, pg 2)
"Civil War Soldiers of Atkinson, NH" (Virtual Cemetery on Find a Grave)
Atkinson Cemetery (Flickr photo album w/ Civil War graves)

Over forty New Hampshire men, whether a native son, a resident, or a volunteer accredited to the state, earned a Medal of Honor for their actions during the Civil War [see tab on left for a full listing]. Here is one story:  

GEORGE PARSONS DOW
​Sergeant, Company C, 7th New Hampshire Infantry.
Place/date of action: Near Richmond, Virginia, October 1864
Date of issue: 10 May 1884
Citation: Gallantry while in command of his company during a reconnaissance toward Richmond. 

George P. Dow of Atkinson, N.H. enlisted as a private on 14 Oct 1861, in neighboring Plaistow, NH. He was mustered in on 6 Nov. in Manchester, and was soon promoted to corporal. On the 18th of July, '62, he was again promoted, to 1st Sergeant. He was mustered out of service on 22 Dec 1864. He had been wounded several times during his three years with the regiment, though not injured bad enough to remove him from active duty.

In October of 1864, while in command of his company, which held the extreme left of the regimental line, the unit became separated from the others while on reconnaissance in the vicinity of Richmond. Upon reaching the enemy breastworks they halted, then realizing the rest of the regiment was not behind them, and called a retreat. Being the first of the Union troops to reach that close to the city, the information gathered was of great importance once they returned to their lines.

George P. Dow, the son of Moses and Sally P (Hanson) Dow, was born in Atkinson, NH on 7 Aug 1840. He was married, on 7 Dec 1865, to Julia A. Carlton [registered in Lawrence, MA vr]. He died on 28 Aug 1910 in Atkinson, aged 70 yrs, and was buried in the town cemetery [NH vr]. They had two children: i. Mary A Dow, b. 21 Sept 1866; marr. William C. Farley ii. ch, b/d 24 May 1880
   
George was a farmer in Atkinson, later becoming the proprietor of a country store in town, and was its postmaster for 26 years. His former homestead still stands, across from the town common named for him.

Links:
Deeds of Valor, pg 436-7
History of the Seventh Regiment: pg 96 (photo), pg 530 (bio)
Muster Out Roll for Company C [family search link]